Time-of-flight mass spectrometry is undergoing enormous technical improvements which make it possible, in principle, to obtain very accurate mass determinations. However, mathematical representation of the calibration curve, that is, the functional relationship between mass and time-of-flight, has not been satisfactorily achieved yet and presents a particular problem.
Time-of-flight mass spectrometers with ionization by matrix-assisted laser desorption (MALDI) are currently advancing, in mass ranges between 1,000 and 6,000 Daltons, to mass resolutions of better than R=m/Δm=50,000, where Δm is the full width at half-maximum (FWHM) of the mass signal at mass m. These values are surprising; they mean that, in the upper mass range, the MALDI time-of-flight mass spectrometers surpass other kinds of mass spectrometers, such as ICR mass spectrometers and Orbitrap™, whose fundamentally high mass resolution decreases as 1/m towards higher masses. The successes are based on improvements to the acceleration electronics and the detector, an increase in the sampling rates of the transient recorders and, in particular, better mastery of the MALDI processes by using improved laser technology as described, for example, in the German patent publication DE 10 2004 044 196 A1 (A. Hase et al., corresponding to GB 2 421 352 A and published U.S. Patent Application 20060071160). A significant contribution to the continuous improvement of this technology has been made by the long known time-delayed acceleration, as described in U.S. Pat. No. 5,654,545, for example, and, in particular, by the shaping of a temporally changing acceleration as described in German patent specification DE 196 38 577 C1 (J. Franzen, corresponding to GB 2 317 495 B and U.S. Pat. No. 5,969,348). In the simplest case, this temporally changing acceleration uses an RC element to slow down the switching on of the acceleration. This causes the region of maximum mass resolution to extend evenly over a wide mass range rather than being located at only one point of the mass spectrum.
Time-of-flight mass spectrometers with orthogonal ion injection (OTOF), which are usually operated with electrospray ion sources (ESI) but now increasingly with other types of ion source as well, are also advancing into these regions of mass resolution by virtue of similar technical improvements. Here, too, acceleration of the ions of a primary ion beam perpendicular to the previous direction, into the flight path of the mass spectrometer is carried out instantaneously by suddenly switching on the accelerating voltage.
It is fundamentally impossible to instantaneously (i.e., in no time), switch on an accelerating field by switching a voltage on a diaphragm which is arranged in a stack of other diaphragms and has a considerable capacitance with respect to the others. If the diaphragm has a low-resistance connection to a power supply, then, once the capacitance has charged up, which takes a finite time, a periodic overshooting always takes place due to the inductance of the supply lead. This overshooting is only slowly damped by the ever-present resistances of the materials, and has very damaging effects on the acceleration of the ions and hence on the calibration curve. The overshooting is therefore damped, as far as possible, by additional resistors in the supply lead to 140 a level where the aperiodic limiting case of the switching occurs, which results in a constant voltage in the shortest time, but not without a transition curve. To permit manufacture with better reproducibility, a slightly larger resistor is used, thus even falling short of this essentially ideal aperiodic limiting case, so that the final strength of the acceleration field is approached in the form of a creeping exponential curve. This “dynamic acceleration” bends the calibration curve in a way that closely resembles a MALDI time-of-flight mass spectrometer.
If ions are accelerated ideally to a kinetic energy E simultaneously and in an infinitesimally short time, one can determine the relationship between their time of flight Δt=t−t0 over a distance L and their mass m from the basic equations:E=(m/2)×v2=(m/2)×L2/(t−t0)2;  [1]m=2E×(t−t0)2/L2;  [2]t=t0+L×√(m/2E)  [3]For various reasons, these equations are only valid as an approximation.
It has been long known that in MALDI mass spectrometers, the ions of all masses obtain a common velocity distribution with a common average initial velocity v0 in the adiabatically expanding plasma of the matrix-assisted laser desorption (MALDI). The kinetic energy E after the electric post-acceleration of the ions thus comprises two components: the energy EU caused by the electric acceleration, and the initial energy E0=(m/2)×v02, which results from the MALDI process:E=EU+E0=EU+(m/2)×v02.  [4]
If one introduces this additional premise into the above Equations [2] and [3] and then introduces some approximations which are based on the fact that the initial energy E0 is very small compared to the energy EU from the electric acceleration, one obtains a very good approximate equation for the time-of-flight as a function of the mass:t≈c0×(√m)0+c1×(√m)1+c3×(√m)3,  [5]and also a very good approximate equation for the mass as a function of the time of flight:m=k2(t−t0)2+k4(t−t0)4,  [6]which can be widely used for both MALDI time-of-flight mass spectrometers and time-of-flight mass spectrometers with orthogonal injection (OTOF-MS). The coefficients c0 to c3 t0, k2 and k4 are determined by mathematical fittings from the ion signals of a mass spectrum of a calibration substance with accurately known masses. Such fitting procedures are familiar to those skilled in the art. For an OTOF-MS, where the ions do not have an initial velocity, the coefficient c3 can even be assumed to be zero. The physical meaning and origin of the coefficients is immaterial for the application, but they are given below for reasons of completeness:c0=t0; c1≈L/√(2EU);c3≈L v02/(√32(√EU)3);k2=2EU/L2+m0v02/L2;k4=2EUv02/L4.  [7]
Since these approximations do not lead to very good mass accuracies, further terms with additional coefficients are usually added, for example the terms c2×(√m)2 and c4×(√m)4 in Equation [5]. The mass accuracy that can be achieved with the equations moderately improves as the number of terms in the expansion series of the equations increases. This is only valid in the upper mass range, however; in the lower mass range, deviations occur which are not improved by these additional terms. The coefficients c2 and C4 cannot be given a physical meaning here.
In time-of-flight mass spectrometers, the ion currents of the ions reaching the detector are amplified, digitized with a constant frequency and stored as digital values in the order they were measured. Normal practice is to acquire many such single spectra in succession from one sample and add them together to form a sum spectrum, digital value by digital value. The original sum mass spectrum therefore includes a long series of digital measurement values where the relevant times of flight t of the ion signals do not appear explicitly, but only form the indices of the measurement series. The measurement series is analyzed for the occurrence of prominent signals; these represent the ion signals. A large number of algorithms and software programs, which are usually called “peak picking programs”, are available for the identification of these ion signals. For an ion peak whose measured values regularly extend over several indices, the time of flight t is interpolated from the indices of the measured values. By using a good peak picking procedure, it is possible to obtain accuracies for the times of flight which are by far better than the time intervals of the digitizing rate.
The accuracy of the time of flight determination depends on the digitizing rate. The transient recorders of contemporary commercial time-of-flight mass spectrometers usually use a digitizing rate of two gigahertz; it is foreseeable, however, that measurement frequencies of eight or ten gigahertz will be available and will be used in the future. It is therefore to be expected that by using good interpolations of the peak picking procedures, accuracies of approximately one hundredth of a nanosecond will be achievable for the time-of-flight determination. A very accurate peak picking procedure based on the simultaneous analysis of all the ion signals of one isotopic group is presented in patent specification DE 198 03 309 C1 (C. Köster, corresponding to U.S. Pat. No. 6,188,064 and GB 2 333 893 B). Since a mass range of up to some 6,000 Daltons is acquired in approximately 100 microseconds, mass accuracies better than one part per million are to be expected in principle, but they have not been achieved as yet.
It is not possible to achieve mass accuracies of better than ten to a hundred parts per million (10 to 100 ppm) of the mass using the Equations [5] or [6] above as calibration functions. Residual errors therefore remain between the values thus obtained and the true values of the masses. If the residual errors are plotted over the mass axis, the diagram shows that, for repeated measurements with the same mass spectrometer, the error curve constantly exhibits similar behavior. These errors are therefore largely systematic residual errors rather than statistical errors. The established method of improving the mass accuracy is thus to approximate the behavior of the error curve using a higher order polynomial, and to use this polynomial to calculate and make allowance for the systematic residual errors with respect to the calibration curve. The skillful application of a seventh order polynomial, for example, in the mass range between 1,000 and 3,000 daltons means that mass accuracies of approximately five parts per million of the mass can be achieved.
The calibration mass spectrum must have a large number of fitting points to determine this polynomial of the systematic residual errors. It is known that a polynomial of the seventh order can be determined with only 8 masses as fitting points, but the polynomial can then assume values between the masses of the fitting points which are at an arbitrary distance. This polynomial method must therefore be applied with great care: with at least around 15 fitting points, which also have to fulfill further conditions, for instance separations which are as evenly spaced as possible with slightly narrower separations at the upper and lower limits. Mixtures of calibration substances that furnish more that 15 fitting points cleanly and without interference from impurity signals are difficult to produce. Moreover, since it has become customary to manually delete outlying measurements during the calibration, there is an exceptionally high risk of two calibrations by two different persons producing widely differing results. Furthermore, the polynomial method does not allow the calibration curve to be used outside the calibrated range, because the values of the polynomial outside the calibrated range usually stray randomly fast and randomly far in unpredictable directions.